arXiv subjects

Clayton G. Webster

Publications and source records attributed to Clayton G. Webster.

16 recordsLinked to original sources

arXiv2024

Examining Policy Entropy of Reinforcement Learning Agents for Personalization Tasks

This effort is focused on examining the behavior of reinforcement learning systems in personalization environments and detailing the differences in policy entropy associated with the type of learning algorithm utilized. We demonstrate that Policy Optimization agents often possess low-entropy policies during training, which in practice results in agents prioritizing certain actions and avoiding others. Conversely, we also show that Q-Learning agents are far less susceptible to such behavior and generally maintain high-entropy policies throughout training, which is often preferable in real-world applications. We provide a wide range of numerical experiments as well as theoretical justification to show that these differences in entropy are due to the type of learning being employed.

cs.LG↗

arXiv2022

An adaptive stochastic gradient-free approach for high-dimensional blackbox optimization

In this work, we propose a novel adaptive stochastic gradient-free (ASGF) approach for solving high-dimensional nonconvex optimization problems based on function evaluations. We employ a directional Gaussian smoothing of the target function that generates a surrogate of the gradient and assists in avoiding bad local optima by utilizing nonlocal information of the loss landscape. Applying a deterministic quadrature scheme results in a massively scalable technique that is sample-efficient and achieves spectral accuracy. At each step we randomly generate the search directions while primarily following the surrogate of the smoothed gradient. This enables exploitation of the gradient direction while maintaining sufficient space exploration, and accelerates convergence towards the global extrema. In addition, we make use of a local approximation of the Lipschitz constant in order to adaptively adjust the values of all hyperparameters, thus removing the careful fine-tuning of current algorithms that is often necessary to be successful when applied to a large class of learning tasks. As such, the ASGF strategy offers significant improvements when solving high-dimensional nonconvex optimization problems when compared to other gradient-free methods (including the so called "evolutionary strategies") as well as iterative approaches that rely on the gradient information of the objective function. We illustrate the improved performance of this method by providing several comparative numerical studies on benchmark global optimization problems and reinforcement learning tasks.

math.OC↗

arXiv2021

Analysis of The Ratio of $\ell_1$ and $\ell_2$ Norms in Compressed Sensing

We first propose a novel criterion that guarantees that an $s$-sparse signal is the local minimizer of the $\ell_1/\ell_2$ objective; our criterion is interpretable and useful in practice. We also give the first uniform recovery condition using a geometric characterization of the null space of the measurement matrix, and show that this condition is easily satisfied for a class of random matrices. We also present analysis on the robustness of the procedure when noise pollutes data. Numerical experiments are provided that compare $\ell_1/\ell_2$ with some other popular non-convex methods in compressed sensing. Finally, we propose a novel initialization approach to accelerate the numerical optimization procedure. We call this initialization approach \emph{support selection}, and we demonstrate that it empirically improves the performance of existing $\ell_1/\ell_2$ algorithms.

math.NA↗

arXiv2020

A nonlocal feature-driven exemplar-based approach for image inpainting

We present a nonlocal variational image completion technique which admits simultaneous inpainting of multiple structures and textures in a unified framework. The recovery of geometric structures is achieved by using general convolution operators as a measure of behavior within an image. These are combined with a nonlocal exemplar-based approach to exploit the self-similarity of an image in the selected feature domains and to ensure the inpainting of textures. We also introduce an anisotropic patch distance metric to allow for better control of the feature selection within an image and present a nonlocal energy functional based on this metric. Finally, we derive an optimization algorithm for the proposed variational model and examine its validity experimentally with various test images.

cs.CV↗

arXiv2019

Closure Learning for Nonlinear Model Reduction Using Deep Residual Neural Network

Developing accurate, efficient, and robust closure models is essential in the construction of reduced order models (ROMs) for realistic nonlinear systems, which generally require drastic ROM mode truncations. We propose a deep residual neural network (ResNet) closure learning framework for ROMs of nonlinear systems. The novel ResNet-ROM framework consists of two steps: (i) In the first step, we use ROM projection to filter the given nonlinear PDE and construct a filtered ROM. This filtered ROM is low-dimensional, but is not closed (because of the PDE nonlinearity). (ii) In the second step, we use ResNet to close the filtered ROM, i.e., to model the interaction between the resolved and unresolved ROM modes. We emphasize that in the new ResNet-ROM framework, data is used only to complement classical physical modeling (i.e., only in the closure modeling component), not to completely replace it. We also note that the new ResNet-ROM is built on general ideas of spatial filtering and deep learning and is independent of (restrictive) phenomenological arguments, e.g., of eddy viscosity type. The numerical experiments for the 1D Burgers equation show that the ResNet-ROM is significantly more accurate than the standard projection ROM. The new ResNet-ROM is also more accurate and significantly more efficient than other modern ROM closure models.

physics.flu-dyn↗

arXiv2019

A Weighted $\ell_1$-Minimization Approach For Wavelet Reconstruction of Signals and Images

In this effort, we propose a convex optimization approach based on weighted $\ell_1$-regularization for reconstructing objects of interest, such as signals or images, that are sparse or compressible in a wavelet basis. We recover the wavelet coefficients associated to the functional representation of the object of interest by solving our proposed optimization problem. We give a specific choice of weights and show numerically that the chosen weights admit efficient recovery of objects of interest from either a set of sub-samples or a noisy version. Our method not only exploits sparsity but also helps promote a particular kind of structured sparsity often exhibited by many signals and images. Furthermore, we illustrate the effectiveness of the proposed convex optimization problem by providing numerical examples using both orthonormal wavelets and a frame of wavelets. We also provide an adaptive choice of weights which is a modification of the iteratively reweighted $\ell_1$-minimization method.

eess.IV↗

arXiv2019

A Polynomial-Based Approach for Architectural Design and Learning with Deep Neural Networks

In this effort we propose a novel approach for reconstructing multivariate functions from training data, by identifying both a suitable network architecture and an initialization using polynomial-based approximations. Training deep neural networks using gradient descent can be interpreted as moving the set of network parameters along the loss landscape in order to minimize the loss functional. The initialization of parameters is important for iterative training methods based on descent. Our procedure produces a network whose initial state is a polynomial representation of the training data. The major advantage of this technique is from this initialized state the network may be improved using standard training procedures. Since the network already approximates the data, training is more likely to produce a set of parameters associated with a desirable local minimum. We provide the details of the theory necessary for constructing such networks and also consider several numerical examples that reveal our approach ultimately produces networks which can be effectively trained from our initialized state to achieve an improved approximation for a large class of target functions.

cs.LG↗

arXiv2018

Non-intrusive inference reduced order model for fluids using linear multistep neural network

In this effort we propose a data-driven learning framework for reduced order modeling of fluid dynamics. Designing accurate and efficient reduced order models for nonlinear fluid dynamic problems is challenging for many practical engineering applications. Classical projection-based model reduction methods generate reduced systems by projecting full-order differential operators into low-dimensional subspaces. However, these techniques usually lead to severe instabilities in the presence of highly nonlinear dynamics, which dramatically deteriorates the accuracy of the reduced-order models. In contrast, our new framework exploits linear multistep networks, based on implicit Adams-Moulton schemes, to construct the reduced system. The advantage is that the method optimally approximates the full order model in the low-dimensional space with a given supervised learning task. Moreover, our approach is non-intrusive, such that it can be applied to other complex nonlinear dynamical systems with sophisticated legacy codes. We demonstrate the performance of our method through the numerical simulation of a two-dimensional flow past a circular cylinder with Reynolds number Re = 100. The results reveal that the new data-driven model is significantly more accurate than standard projection-based approaches.

physics.comp-ph↗

arXiv2017

On the Lebesgue Constant of Weighted Leja Points for Lagrange Interpolation on Unbounded Domains

This work focuses on weighted Lagrange interpolation on an unbounded domain, and analyzes the Lebesgue constant for a sequence of weighted Leja points. The standard Leja points are a nested sequence of points defined on a compact subset of the real line, and can be extended to unbounded domains with the introduction of a weight function $w:\mathbb{R}\rightarrow [0,1]$. Due to a simple recursive formulation in one dimension, such abscissas provide a foundation for high-dimensional approximation methods such as sparse grid collocation, deterministic least squares, and compressed sensing. Just as in the unweighted case of interpolation on a compact domain, we use results from potential theory to prove that the Lebesgue constant for the Leja points grows subexponentially with the number of interpolation nodes.

math.NA↗

arXiv2017

Compressed sensing approaches for polynomial approximation of high-dimensional functions

In recent years, the use of sparse recovery techniques in the approximation of high-dimensional functions has garnered increasing interest. In this work we present a survey of recent progress in this emerging topic. Our main focus is on the computation of polynomial approximations of high-dimensional functions on $d$-dimensional hypercubes. We show that smooth, multivariate functions possess expansions in orthogonal polynomial bases that are not only approximately sparse, but possess a particular type of structured sparsity defined by so-called lower sets. This structure can be exploited via the use of weighted $\ell^1$ minimization techniques, and, as we demonstrate, doing so leads to sample complexity estimates that are at most logarithmically dependent on the dimension $d$. Hence the curse of dimensionality - the bane of high-dimensional approximation - is mitigated to a significant extent. We also discuss several practical issues, including unknown noise (due to truncation or numerical error), and highlight a number of open problems and challenges.

math.NA↗

arXiv2016

Polynomial approximation via compressed sensing of high-dimensional functions on lower sets

This work proposes and analyzes a compressed sensing approach to polynomial approximation of complex-valued functions in high dimensions. Of particular interest is the setting where the target function is smooth, characterized by a rapidly decaying orthonormal expansion, whose most important terms are captured by a lower (or downward closed) set. By exploiting this fact, we present an innovative weighted $\ell_1$ minimization procedure with a precise choice of weights, and a new iterative hard thresholding method, for imposing the downward closed preference. Theoretical results reveal that our computational approaches possess a provably reduced sample complexity compared to existing compressed sensing techniques presented in the literature. In addition, the recovery of the corresponding best approximation using these methods is established through an improved bound for the restricted isometry property. Our analysis represents an extension of the approach for Hadamard matrices in [5] to the general case of continuous bounded orthonormal systems, quantifies the dependence of sample complexity on the successful recovery probability, and provides an estimate on the number of measurements with explicit constants. Numerical examples are provided to support the theoretical results and demonstrate the computational efficiency of the novel weighted $\ell_1$ minimization strategy.

math.NA↗

arXiv2015

Analysis of quasi-optimal polynomial approximations for parameterized PDEs with deterministic and stochastic coefficients

In this work, we present a generalized methodology for analyzing the convergence of quasi-optimal Taylor and Legendre approximations, applicable to a wide class of parameterized elliptic PDEs with finite-dimensional deterministic and stochastic inputs. Such methods construct an optimal index set that corresponds to the sharp estimates of the polynomial coefficients. Our analysis furthermore represents a new approach for estimating best $M$-term approximation errors by means of coefficient bounds, without using Stechkin inequality. The framework we propose for analyzing asymptotic truncation errors is based on an extension of the underlying multi-index set into a continuous domain, and then an approximation of the cardinality (number of integer multi-indices) by its Lebesgue measure. Several types of isotropic and anisotropic (weighted) multi-index sets are explored, and rigorous proofs reveal sharp asymptotic error estimates in which we achieve sub-exponential convergence rates of the form $M \exp(-(κM)^{1/N})$, with $κ$ a constant depending on the shape and size of multi-index sets) with respect to the total number of degrees of freedom. Through several theoretical examples, we explicitly derive the constant $κ$ and use the resulting sharp bounds to illustrate the effectiveness of Legendre over Taylor approximations, as well as compare our rates of convergence with current published results. Computational evidence complements the theory and shows the advantage of our generalized framework compared to previously developed estimates.

math.AP↗

arXiv2015

A sparse grid method for Bayesian uncertainty quantification with application to large eddy simulation turbulence models

There is wide agreement that the accuracy of turbulence models suffer from their sensitivity with respect to physical input data, the uncertainties of user-elected parameters, as well as the model inadequacy. However, the application of Bayesian inference to systematically quantify the uncertainties in parameters, by means of exploring posterior probability density functions (PPDFs), has been hindered by the prohibitively daunting computational cost associated with the large number of model executions, in addition to daunting computation time per one turbulence simulation. In this effort, we perform in this paper an adaptive hierarchical sparse grid surrogate modeling approach to Bayesian inference of large eddy simulation (LES). First, an adaptive hierarchical sparse grid surrogate for the output of forward models is constructed using a relatively small number of model executions. Using such surrogate, the likelihood function can be rapidly evaluated at any point in the parameter space without simulating the computationally expensive LES model. This method is essentially similar to those developed in [62] for geophysical and groundwater models, but is adjusted and applied here for a much more challenging problem of uncertainty quantification of turbulence models. Through a numerical demonstration of the Smagorinsky model of two-dimensional flow around a cylinder at sub-critical Reynolds number, our approach is proven to significantly reduce the number of costly LES executions without losing much accuracy in the posterior probability estimation. Here, the model parameters are calibrated against synthetic data related to the mean flow velocity and Reynolds stresses at different locations in the flow wake. The influence of the user-elected LES parameters on the quality of output data will be discussed.

math.NA↗

arXiv2015

A Dynamically Adaptive Sparse Grid Method for Quasi-Optimal Interpolation of Multidimensional Analytic Functions

In this work we develop a dynamically adaptive sparse grids (SG) method for quasi-optimal interpolation of multidimensional analytic functions defined over a product of one dimensional bounded domains. The goal of such approach is to construct an interpolant in space that corresponds to the "best $M$-terms" based on sharp a priori estimate of polynomial coefficients. In the past, SG methods have been successful in achieving this, with a traditional construction that relies on the solution to a Knapsack problem: only the most profitable hierarchical surpluses are added to the SG. However, this approach requires additional sharp estimates related to the size of the analytic region and the norm of the interpolation operator, i.e., the Lebesgue constant. Instead, we present an iterative SG procedure that adaptively refines an estimate of the region and accounts for the effects of the Lebesgue constant. Our approach does not require any a priori knowledge of the analyticity or operator norm, is easily generalized to both affine and non-affine analytic functions, and can be applied to sparse grids build from one dimensional rules with arbitrary growth of the number of nodes. In several numerical examples, we utilize our dynamically adaptive SG to interpolate quantities of interest related to the solutions of parametrized elliptic and hyperbolic PDEs, and compare the performance of our quasi-optimal interpolant to several alternative SG schemes.

math.NA↗

arXiv2015

Accelerating stochastic collocation methods for partial differential equations with random input data

This work proposes and analyzes a generalized acceleration technique for decreasing the computational complexity of using stochastic collocation (SC) methods to solve partial differential equations (PDEs) with random input data. The SC approaches considered in this effort consist of sequentially constructed multi-dimensional Lagrange interpolant in the random parametric domain, formulated by collocating on a set of points so that the resulting approximation is defined in a hierarchical sequence of polynomial spaces of increasing fidelity. Our acceleration approach exploits the construction of the SC interpolant to accelerate the underlying ensemble of deterministic solutions. Specifically, we predict the solution of the parametrized PDE at each collocation point on the current level of the SC approximation by evaluating each sample with a previously assembled lower fidelity interpolant, and then use such predictions to provide deterministic (linear or nonlinear) iterative solvers with improved initial approximations. As a concrete example, we develop our approach in the context of SC approaches that employ sparse tensor products of globally defined Lagrange polynomials on nested one-dimensional Clenshaw-Curtis abscissas. This work also provides a rigorous computational complexity analysis of the resulting fully discrete sparse grid SC approximation, with and without acceleration, which demonstrates the effectiveness of our proposed methodology in reducing the total number of iterations of a conjugate gradient solution of the finite element systems at each collocation point. Numerical examples include both linear and nonlinear parametrized PDEs, which are used to illustrate the theoretical results and the improved efficiency of this technique compared with several others.

math.NA↗

arXiv2014

A Multilevel Stochastic Collocation Method for Partial Differential Equations with Random Input Data

Stochastic collocation methods for approximating the solution of partial differential equations with random input data (e.g., coefficients and forcing terms) suffer from the curse of dimensionality whereby increases in the stochastic dimension cause an explosion of the computational effort. We propose and analyze a multilevel version of the stochastic collocation method that, as is the case for multilevel Monte Carlo (MLMC) methods, uses hierarchies of spatial approximations to reduce the overall computational complexity. In addition, our proposed approach utilizes, for approximation in stochastic space, a sequence of multi-dimensional interpolants of increasing fidelity which can then be used for approximating statistics of the solution as well as for building high-order surrogates featuring faster convergence rates. A rigorous convergence and computational cost analysis of the new multilevel stochastic collocation method is provided, demonstrating its advantages compared to standard single-level stochastic collocation approximations as well as MLMC methods. Numerical results are provided that illustrate the theory and the effectiveness of the new multilevel method.

math.NA↗